Mesoscopic samples contain a large number of atoms but are small on the scale

of a temperature-dependent ”coherence length”. On such scales electronic and

mechanical phenomena coexist:Mesoscopic Nanoelectromechanics

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Quantum Coherence of Electrons

•Spatial quantization of electronic motion

•Quantum tunneling of electrons

•Resonance transmission phenomenon

•Tunnel charge relaxation and tunnel resistance

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Spatial quantization of orbital motion

•

For a sample with symmetric shape the electronic spectrum is degenerate

•

A distortion of the geometrical shape tends to lift degeneracies.

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Quantum Level Spacing

Estimation

of

average

level

spacing,

assuming

all

quantum

states

are

nondegenerate

and

homogeneously

distributed

in

energy

N

–

total number of

electrons

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Quantum Tunneling

The

classically

moving

electron

is

reflected

by

a

potential

barrier

and

can

not

be

“seen”

in

the

region

x

>

0.

The

quantum

particle

can

penetrate

into

such

a

forbidden

region.

Under-the-barrier propagation:

Under-the-barrier propagation is called

tunneling. Wave function’s decay length

is called thetunneling length.

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Tunneling through a Barrier

Due to quantum tunneling a particle has a finite

probabilitytopenetrate

through abarrier

of arbitrary height.

t

andr

are probability amplitudes for thetransmission

andreflection

ofthe particle. These parameters characterise the barrier and can oftenbe considered to be only weakly energy dependent.

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Tunneling Width of a Quantum Level

LetN

be the number of ”tries” madebefore the particle finally escapes the dot:

Escape

time

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Resonant Tunneling

Electronic waves, like ordinary waves, experience a set of multiple reflections as theymove back and forth between two barriers. The total probability amplitude for the transferof a particle can be viewed as a sum of amplitudes, each corresponding to escape afteran increasing number of “bounces” between the barriers.

Ifp = pn= nh/2dwe have D=1independently

of the barrier transparency! (Resonance)

Breit-Wigner formula

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Tunneling Resistance

An

electric fieldmust be present in the vicinity of the barrier in order tocompensate

for the ”scattering force” of the potential barrier and achieve astationary current flow

Solve this for times longer than the damping time for the beam by expansion

in terms of eigenfunctions:

The equation for theexpansion coefficientsan

is

Driven Damped Beams

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Using the definitions of the eigenfunctions and their properties, and the

definition of the complex-valuedeigenfrequencies

’n

this can be written as:

For捬獥1, only then=1

term has asignificant amplitude,

given by:

For a uniform force distribution,f(x)=f0,the integral is evaluated to1L2,

1=0.8309and we have, since’n=(1-i/Q)n:

Lecture2: Electronics and Mechanics on the Nanometer Scale

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The displacement of a forced damped beam driven near its fundamental

frequence is–

as we have seen–

given by

In the absence of noise the motion is purely harmonic at the carrier frequency

.䉵ihe牥i猠di獳spain⡦inie

Q), there is also necessarilynoise

and a

noise forcefN(t)

that can be expanded in terms of the eigenfunctionsun(x):

As we discussed already dissipation drives the beam to equilibrium with its

environment at temperatureT

and the stochastic noise force maintains theequilibrium.

Dissipation-Induced Amplitude Noise

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Without driving force the mean total energy for each mode iskBT.This requires

the spectral density of the noise forcefN,n(t)

to be:

Force per length, hence the

termL2, which is not there

for a simple harmonic osc.

Using this result we can calculate the spectral density for the thermally driven

amplitude as

Lecture2: Electronics and Mechanics on the Nanometer Scale

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Lecture2: Electronics and Mechanics on the Nanometer Scale

Speaker: ProfessorRobertShekhter, Gothenburg University2009

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Comments to the next slide

This equation can be used to find thevibrational spectrum

of adouble clamped beam. Inserting an inertion term andextractind an external force we find thye equation. Notethat it differs from the wave equation due to fourth orderspacial derivative instead second one is present. Theboundary conditions just demand thatdiscplacement

anddeformation

of a beam material are equal tozero

if end ofthebeam

are spacialyfixed.

Discrete sets of different solutions(modes) are presentedhere. Notice thatfrequency

isinversely proportional

to thesquare

of the beamlength.

(This is in contrast to the bulk elastic vibrations whichlowers phjononic frequency is inversely proportionasl to thelewngth of the sample not to the length squared.)

Lecture2: Electronics and Mechanics on the Nanometer Scale

Speaker: ProfessorRobertShekhter, Gothenburg University2009

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Coments to the next slide

53

The same for the beam clamped only from one side. The boundary condition for

the free side express an absence of the tension and share tension (correct ?) at the

free end. Ther same properties of thye solutions

Coments to the next slide

54

An estimation of the frequency of the nanovibrations.

What is the meaning of the note ”not harmonic”?

Coments to the next slide

55

Would be nice to get comments to ”W” and ”G” which appear on the slide